行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}@{}}1 & {-}2 & 1 & {-}2 \\ 2 & 2 & {-}1 & 0 \\ {-}2 & {-}2 & 2 & {-}1 \\ 0 & 2 & {-}2 & 0\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}2 & 1 & {-}2 \\ 2 & 2 & {-}1 & 0 \\ {-}2 & {-}2 & 2 & {-}1 \\ 0 & 2 & {-}2 & 0\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-2, 0, -1, 0]

\( \qquad\quad = (-1)^{4-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}1 \\ {-}2 & {-}2 & 2 \\ 0 & 2 & {-}2\end{array}\,\right| + (-1)^{4-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}2 & 1 \\ 2 & 2 & {-}1 \\ 0 & 2 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-2)\times(-4) + (-1)^{4-3}\times(-1)\times(-6) \)

\( \qquad\quad = -14 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}1 \\ {-}2 & {-}2 & 2 \\ 0 & 2 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, 2, -2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}2 \\ 0 & 2\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 0 & 2\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ {-}2 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times(-4) + (-1)^{3-2}\times2\times4 + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = -4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}2 & {-}2 \\ 0 & 2\end{array}\,\right| = (-2)\times2 - (-2)\times0 = -4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 0 & 2\end{array}\,\right| = 2\times2 - 2\times0 = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ {-}2 & {-}2\end{array}\,\right| = 2\times(-2) - 2\times(-2) = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & {-}2 & 1 \\ 2 & 2 & {-}1 \\ 0 & 2 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [1, -1, -2]

\( \qquad\quad = (-1)^{3-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 0 & 2\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & 2\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 2 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times1\times4 + (-1)^{3-2}\times(-1)\times2 + (-1)^{3-3}\times(-2)\times6 \)

\( \qquad\quad = -6 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 0 & 2\end{array}\,\right| = 2\times2 - 2\times0 = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 0 & 2\end{array}\,\right| = 1\times2 - (-2)\times0 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & {-}2 \\ 2 & 2\end{array}\,\right| = 1\times2 - (-2)\times2 = 6 \)